Lg(x^3)-lg(x+4)=lg(2x^2)-lg(2x-4)

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To solve this logarithmic equation, we will first simplify it using logarithmic properties and then solve for x.

Given: lg(x^3) - lg(x+4) = lg(2x^2) - lg(2x-4)

Apply the quotient rule for logarithms to simplify:

lg(x^3) - lg(x+4) = lg(2x^2) - lg(2x-4)

lg(x^3) - lg(x+4) = lg(2x^2/(2x-4))

Now, apply the power rule to simplify further:

3lg(x) - lg(x+4) = lg(x^2)/(x-2)

Next, apply the product rule to simplify:

3lg(x) - lg(x+4) = 2lg(x) - lg(x-4)

Now, combine like terms:

3lg(x) - lg(x+4) - 2lg(x) + lg(x-4) = 0

lg(x) - lg(x+4) + lg(x-4) = 0

Now, use the properties of logarithms:

lg[(x(x-4))/(x+4)] = 0

Now, remove the logarithm using the property that lg(a) = b is equivalent to a = 10^b:

(x(x-4))/(x+4) = 1

x(x-4) = x + 4

x^2 - 4x = x + 4

x^2 - 5x - 4 = 0

Now, we have a quadratic equation that we can solve to find the values of x. We can factor the equation or use the quadratic formula to find the solutions. By factoring, we get:

(x - 4)(x + 1) = 0

So, the solutions for x are x = 4 and x = -1.